'We've created some feminists!' ... A study group on the Demand the Impossible course at Goldsmith's. Photograph: Frank Baron for the Guardian

A dozen teenagers push their way through the doors of the New Cross branch of Sainsbury's in south-east London, and bear down on the self-service checkouts. To the shoppers filling trolleys in the fresh fruit aisle, they might for a moment look like looters. But then they start yelling.

"Shoppers of Sainsbury's," shouts one woman in the crowd.

"SHOPPERS OF SAINSBURY'S," repeat her friends.

"We are here today."

"WE ARE HERE TODAY."

"In solidarity with Sainsbury's workers."

"IN SOLIDARITY WITH SAINSBURY'S WORKERS."

In this gospel fashion, the teenagers make the incongruous announcement that, despite sizeable profits, the supermarket still fails to pay its workers the London living wage of £8.30 an hour. "Sainsbury's pay for workers is worse than Tesco!" they cry – but by this point said workers are gently ushering the protesters from the shop.

"We are in solidarity with you," they add, but sadly the feeling is not yet mutual.

Just a few days ago, most of these sixth-formers – mainly women, mainly Muslim and working-class – were not particularly political, let alone radical. "I didn't realise I could do these things," says 17-year–old Mamataj Begum, who found herself shaking with emotion afterwards: "I wouldn't have done it this time last week."

But that was before she signed up for Demand the Impossible – a free, five-day course at Goldsmith's College that introduced around 25 teenagers from east and north London to activism and radical politics. By the time I arrive towards the end of the week, the class has had workshops on the Egyptian revolution, Palestine, oil spillages in Africa, and feminism. Prominent leftwinger Mark Fisher has given a talk on the problems with capitalism, and the group has brainstormed some alternatives: social democracy, mutualism, anarchism. They have tried different kinds of activism – leafleting, petitions, and the flashmob – and been encouraged to critique the effectiveness of each tactic. Today, they are planning their own campaigns.

One group wants to set up a support hotline for Asian victims of domestic violence. Another wants to combat negative stereotypes of young people by making films of inspirational teenagers. Still others want to campaign for gender-neutral toys.

"Girls shouldn't have to play with Barbie," says sixth-former Jessica Luong, as her group discusses tactics. "They can play with Action Man too." She would start smalll at first, and focus on individual outlets. What would her protest look like? Leafleting at first, she says. Isn't that a bit boring, asks Holly Rigby, a full-time activist helping to run the course. All right, says Luong: what about a protest outside the store? "Or how about," says Holly, "a protest inside the store!"

Demand the Impossible is in good company. It's one of several leftwing activist workshops to have emerged in the past year. There's UK Feminista, which runs bootcamps for campaigners for gender equality. Netroots hosts an annual conference for online activists. Hard-left group Counterfire organised a citizen activism day at the School of Oriental and African Studies in central London this spring. And then there were the student occupations of 2010, and last year's Occupy camps – all of which were loose attempts to both critique capitalism and engage people in direct action.

But Demand the Impossible – a riff on a quote by Che Guevara – is doing things a bit differently. Its contemporaries are either aimed at more experienced thinkers – in the case of UK Feminista – or run by groups (Counterfire) with explicit, didactic agendas. Some, like Netroots, are niche, while others – Occupy – are notoriously vague. All of them are often criticised for their reliance on middle-class participants versed in protest technique and leftwing theory. During the riots last year, some well-intentioned lefties privately wished they could harness the anger of the looters – the fabled proletariat! – and turn it towards more constructive political protest. But they had little means or history of doing so. Demand the Impossible is an attempt to do what its forebears couldn't: to engage with ordinary, working-class, predominantly Asian teenagers with little or no experience of activism or leftwing thought. "We went out of our way to find people who weren't particularly radical," says Jacob Mukherjee, the 29-year-old who created the course with his best friend from university, Ed Lewis. Both teachers at north London state schools, they asked their students to spread the word about the summer school among their friends. Many of their students ended up enlisting themselves – and the pair accepted applications from anyone with even a vague idea of what activism was. "The bar was very low," says Lewis. "One said their teacher had gone on strike – and that was enough."

At lunchtime, someone picks up my sandwich by mistake, and there is a scramble to find who it is. "What did we say about the difference between theft and disappropriation?" laughs Lewis, in a tongue-in-cheek reference to Marxist theory. It's a self-effacing reminder that, though this is a course run by lefties, they don't take themselves or their politics as seriously as cynics might think. "We both recognise the problems with the left," Lewis later admits. "He's not teaching his opinions," says pink-haired, 16-year-old Eloise Hart, of Mukherjee. "He's teaching you to know what your opinions are – and how to do something about them."

The course is structured through the prism of, in Lewis's words, "transcending capitalism" – but I don't meet anyone who thinks they have been brainwashed. "The course isn't actually about being a capitalist or a socialist," argues Luong. "It's not about labels. It's about exposing yourself to different ideas."

At one point, some of the students are sent to ask the people of New Cross to sign a petition that, amusingly, calls for the end of capitalism. It garners 11 signatures in half an hour – perhaps not enough to bring about a global revolution. Then again, the petition isn't a serious attempt to radicalise either New Cross or the people on the course. It is just a useful means of sparking a discussion about the effectiveness of different activist tactics. It's a means of evaluation, not indoctrination, say Mukherjee and Lewisnot indoctrination.

"They haven't indoctrinated me," smiles Begum. "At the end of the week, I'm still a little bit capitalist."

If anything, Lewis and Mukherjee are surprised by how wedded their students remain to capitalist ideas, and to the virtues of meritocracy. They are interested in critiquing capitalism, sure – but most of them don't yet have fundamental problems with it. Early on in the week, the pair ask the students to come up with a fairer way of structuring society. The world, they argue, is currently shaped a bit like a pyramid – with the rich few in the tip at the top, and the impoverished masses in the wide bit at the bottom. What shape, they ask, would create a fairer system? A couple of people suggest a flat line, putting everyone on the same level. But most wanted a less radical redesign. A semi-circle would do – more room at the top, but still some opportunity for people to work their way up the social ladder. "They all watch Alan Sugar," says Mukherjee. "So they all believe: if I've got enough drive, I can make it. They don't really see anyone from other social spheres. So they don't really know how other people are so much more advantaged than them."

Still, he shouldn't be despondent: "They've taught us to think about things in a different way," says Begum. "You can't really do that in lessons."

Ah, the national curriculum – another recurrent theme. If the course is biased, say several students, it's no more so than their human science A-levels at school, which rarely encourage independent thought, and hardly ever challenge capitalism. "In geography, you do a tiny bit of economics, but you don't learn about the pros and cons of capitalism," says Luong. "In history, you don't learn about radical politics. You do explore ideas, but only in the way that the curriculum teaches them." In two years of business studies, 18-year-old Rosemary Ovensehi says she was never prompted to question whether businesses should care about anything other than profit. But on the first day at Demand the Impossible, she learned about Shell's chequered environmental record in Nigeria – and for the first time realised profit isn't the only thing a business should be forced to worry about.

The week is full of such Damascene moments. As a child, Adenike Ijanusi, now 17, had always wanted to be a fireman, so a few years ago she did work experience at a fire station. When they arrived, all the workies were lined up and asked to introduce themselves. But when they came to Ijanusi, the firemen missed her out. Then they were measured for firemen's boots – all except Ijanusi. As the only girl there, she was shunned. At the time, she just thought sexism was a fact of life, and set her sights on other goals: "I just washed it out of my mind." But the course, with its side-focus on feminism, made her realise sexism is something that can and should be fought.

"We've created some feminists!" says a delighted Maeve McKeown, a PhD candidate at UCL who gave an introduction to feminism on day two. "There were a few who were feminists at the start, but I don't think most of them really understood how it extended into their own lives. That's the difference to how it's been taught in their schools. There, it's just about the suffragettes."

"I wasn't really a feminist before," says Begum. "I thought it was about fighting for the right to vote. Now I realise how relevant it is, and how it's about domestic violence and street harassment." Some girls were so moved, they now want to set up a support hotline for domestic violence victims from ethnic backgrounds. For her part, the course has helped Begum develop ideas about the problematic way her cultural community (as opposed to her religion, she argues) treats women. She has noticed that though her culture criticises the revealing nature of some western clothes, "their [own] clothes still show part of their body." Even more problematically: "They think men should speak over women.

At the end of the week, the students give presentations on what they have learned and what related projects they plan to do. On the walls you can still see the massive mind-maps they drew at the beginning of the week – hundreds of scribbled statements of what they thought politics was about: simplistic slogans such as "No War!", "More Taxes!", "End Poverty", "Fair Trade", and "No more having the ability to buy education." But just five days later, those same teenagers are talking confidently about eco-anarchism and mutualism in the same breath as the patriarchy and Mark Fisher. It's hard not to feel a little moved – a feeling that makes Lewis simultaneously elated and crestfallen.

"I was just thinking about going back to school," he says, "and feeling very depressed."